From Chapter VII of [1]} we have the following. Let \(\mathcal {O}_n\)  be the full subcategory of
\([n]/ \Delta \)  whose objects are the surjections \(\phi \colon [n]\twoheadrightarrow [m]\) , for \(m<n\) .
Ordinal sum (on the right) with \([0]\)  defines a functor \(\sigma _0\colon \mathcal {O}_n\rightarrow \mathcal {O}_{n+1}\) . Consider the image
\(\sigma _0(\mathcal {O}_n)\) . The objects of this category consist of all surjections
\(\phi \colon [n+1]\twoheadrightarrow [m]\)  with \(0<m<n+1\)  and with
\(\phi (n)<\phi (n+1)\) . Likewise the morphisms are all morphisms in \([n+1]/ \Delta \)  of the form \(f\colon [m]\rightarrow [k]\)  such that
\(f(m-1)\le k-1\) .
